About physics and consistency of mathematical theo

Posted by: Adib Ben Jebara

About physics and consistency of mathematical theo - 07/19/08 08:04 AM

About physics and consistency of mathematical theories

Can physics make us know that mathematical theories are consistent ? Can
we consider as consistent the mathematical theories which are applied to
physics ?
It seems to me that consistency can be checked with the existence of a
thing satisfying what is assumed.
For instance in set theory, we look for a model to prove the consistency
of axioms.
The physical world satisfies laws and is not unlike a model.

Mr Andreas Blass wrote me that a contradiction could be found far down
the way and a consistency not confirmed by a proof is not mathematical
knowledge.
I think it is unlikely to meet a contradiction far down the way so for
practical purposes the applied theory can be considered consistent.
The fear of a contradiction expressed by some mathematicians is not
reasonable.
Most mathematical theories should be applied to physics in the far future.
Adib Ben Jebara.
Posted by: Zephir

Re: About physics and consistency of mathematical theo - 07/19/08 11:27 PM

Originally Posted By: Adib Ben Jebara
..Can physics make us know that mathematical theories are consistent..
If some math theories are using the same set of postulates (for example the axioms of Peano algebra) and all steps in their derivations are checked by formal proof, then these theories will remain internally consistent and no physics is necessary for verification of their consistency.
Posted by: Adib Ben Jebara

Re: About physics and consistency of mathematical theo - 07/20/08 10:44 AM

Some mathematicians look for a way to be sure that a contradiction will never arise whatever the developments.
Adib Ben Jebara.
Posted by: Zephir

Re: About physics and consistency of mathematical theo - 07/20/08 12:00 PM

Originally Posted By: Adib Ben Jebara
...Some mathematicians look for a way to be sure that a contradiction will never arise whatever the developments...
This is virtually impossible due the Gödel's theorem. We can be never sure by anything, even in highly abstract math. The Aether model explain this by concept of density fluctuations:



These fluctuations can be fully described by math only if we consider the infinite recursion and they're never fully sharp, exact description of reality. Every logical explanation bring a new assumptions on the background. This is because even the most abstract math relies on the concept of countable entities/units, aka numbers which are behaving like colliding particles, thus fulfilling the Fermi statistics.

For example the ripples at the water surface are existing too, but they're not countable, they can penetrate freely - so that the deterministic math has nothing to say about it. It means, whole large area of reality cannot be described by using of deterministic math - while still real and perfectly observable, they're more imaginary, then the most abstract math at all!
Posted by: Adib Ben Jebara

Re: About physics and consistency of mathematical theo - 07/21/08 12:12 AM

There are really proofs of consistency in set theory.
Here, you most probably relate things which are not to be related this way (math and ripples) and what you write about countable entities is not clear.
Adib Ben Jebara.
Posted by: Zephir

Re: About physics and consistency of mathematical theo - 07/21/08 09:09 AM

Originally Posted By: Adib Ben Jebara
what you write about countable entities is not clear
The ripples on the water surface cannot be counted, because they can penetrate each other like ghosts.



Can such objects become a members of some set? You can tell me - I'm not expert in set theory.
Posted by: TheFallibleFiend

Re: About physics and consistency of mathematical - 07/21/08 02:18 PM

Originally Posted By: Adib Ben Jebara
...Some mathematicians look for a way to be sure that a contradiction will never arise whatever the developments...


Mathematicians like a minimum set of primitives from which the rest can be derived. Ideally the primitives are so obvious that no reasonable person could contest them. OTOH, there are cases where a primitive is assumed, but is not entirely logical. Playfair's axiom, for example, is a case in point. Mathematicians tried for centuries to derive it from the other axioms, to no avail. Then some mathematicians questioned "What happens if we make another assumption" and completely consistent non-euclidean geometries were born.

Originally Posted By: Zephir
This is virtually impossible due the Gödel's theorem.


Gödel's theorem does not preclude consistent mathematics. It states that in a sufficiently powerful consistent mathematical system, there are true statements that can't be proven. Mathematics is built axiomatically from a set of primitives and rules which, given certain statements that are acknowledged as true, produce other statements that must be true. (In computer grammars they refer to these as production rules.) There is an alternate statement of it which, IIRC, is something like "Any *complete* mathematical system cannot be consistent."

Posted by: Adib Ben Jebara

Re: About physics and consistency of mathematical - 07/21/08 11:28 PM

To Zephir,
The rippples are in finite number.
Countable in mathematics means in bijection with the set of integers, so infinite.
The set of ripples can be defined but is difficult to handle.
To ThefallibleFiend,
Thank you for your reply.
The last sentence is questionable.
Adib Ben Jebara.
Posted by: Adib Ben Jebara

Re: About physics and consistency of mathematical - 07/22/08 11:05 PM

"Any "complete" mathematical system which includes at least arithmetic cannot be consistent ("small" systems can).
Adib Ben Jebara.